Chebyshev's bias against splitting and principal primes in global fields
Chebyshev's bias against splitting and principal primes in global fields
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DOI：
10.1016/j.jnt.2022.10.005
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发表时间：
2022-03
影响因子：
0.7
通讯作者：
Miho Aoki;S. Koyama
中科院分区：
文献类型：
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作者：
Miho Aoki;S. Koyama
分析了切比雪夫偏向产生的原因。深黎曼假设(DRH)使我们能够揭示，偏差是对整个素数序列进行良好平衡配置的自然现象，在这个意义上，欧拉积收敛于中心。借助于素数的加权计数函数，我们成功地在DRH假设下用一个渐近公式来表示挠度的大小，这给出了切比雪夫偏差的一个新的公式；对于全局场的任意伽罗瓦扩张和伽罗瓦群中的任意元素σ，我们在DRH假设下建立了Frobenius元等于σ的素数偏差的判据.作为应用，我们得到了非分裂非主素数在DRH下的阿贝尔扩张中的倾向性。在正特征情形下，证明了DRH，并且这些结果都无条件成立。
A reason for the emergence of Chebyshev's bias is investigated. The Deep Riemann Hypothesis (DRH) enables us to reveal that the bias is a natural phenomenon for making a well-balanced disposition of the whole sequence of primes, in the sense that the Euler product converges at the center. By means of a weighted counting function of primes, we succeed in expressing magnitudes of the deflection by a certain asymptotic formula under the assumption of DRH, which gives a new formulation of Chebyshev's bias.For any Galois extension of global fields and for any elementσin the Galois group, we establish a criterion of the bias of primes whose Frobenius elements are equal toσunder the assumption of DRH. As an application we obtain a bias toward non-splitting and non-principle primes in abelian extensions under DRH. In positive characteristic cases, DRH is proved, and all these results hold unconditionally.